Here I'll go through a more physical way of viewing Bessel
functions. Bessel functions occur often in the study of
problems with cylindrical symmetry. So when you see cylindrical
symmetry think ``Bessel functions", spherical symmetry think ``Legendre
Polynomials" and when you see Cartesians think ``sine and cosine".

Suppose we want to solve in cylindrical coordinates.
Write . This substitution, a la separation
of variables, leads to the equations

and

The in the last equation is just 2 dimensional, different
from the original used above which is three dimensional.
Eqn. (2) is often refered to as Helmoltz's equation.
To solve it we could use two methods. The first is to seperate variables
into polar coordinates . This gives

which has solutions where n is
an integer. This is the same as Boas chapter 13 equation (5.6).
The equation for R, Boas(5.7) is

We'd like to know how to solve this equation, which is closely
related to Bessel's equation. We don't know how to solve it
so we have two choices. One is to do a power series expansion
as is done in chapter 12 of Boas. Instead we can backtrack to
eqn. (2) and solve it in Cartesian coordinates.
Doing separation of variables again with ,
we obtain

with as a condition
on the two constants and that is obtained
when you go through separation of variables. So F(x,y) can be
written in a rather nice form:

So the general solution can be written

The physical interpretation of this is as follows.
is a plane wave travelling in the
direction. Its magnitude is restricted to be K.
So the general solution to eqn. (2) is
the sum of plane waves all with the same wavelength (or
wave-vector), travelling in any arbitrary direction.
The coefficient indicates the amplitude and
phase of a wave travelling in the direction of .

Since there are a continuous range of angles that the
wave could go in, we should actually write eqn. (7)
as an integral over all possible angles. So writing
and
we can rewrite eqn. (7) as

Here is the angle the is pointing relative
to the x-axis. Letting and
noticing that the integrand is periodic, we can rewrite
this as

This is the general solution to the two dimensional Helmoltz
equation.

Now how do we relate this to eqn. (4) above?
This was obtained by saying we wanted a special solution
that looked like

So we look for solutions to
eqn. (9) which are of this form. That is we
have to hunt for the appropriate . Its not
impossible to see that does
the trick! This gives

Well this does indeed seem to have separated out the
r and components into the desired form. So comparing with
eqn. (10) we see that

This integral will be defined to be equal to a special function.
We'll call it . The is just a pesty
normalization factor that we must include but is quite
uninteresting. The big news is the thing . This is called
a ``Bessel function of the first kind and order n''. The
above integral is an integral representation of that
function. And this
by construction is a solution to eqn. (4). There
is a closely related form to the above integral. Let
. Then

By noting that we have

In summary, Bessel functions can be thought of as the sum of two
dimensional plane waves going in all possible directions.